Normal Distribution Calculator – Find Probability & Percentile

Calculate probabilities and percentiles for a normal distribution with our free online calculator. Input mean and standard deviation to find area under the bell curve.

Mean (μ)

Standard Deviation (σ)

Examples:

X Value

Find P(X < x) - the probability that a value is less than x

Understanding the Normal Distribution

The normal distribution, also called the Gaussian distribution or "bell curve," is the most important probability distribution in statistics. It describes how values cluster around a central mean, with fewer values appearing as you move away from the center.

Many natural phenomena follow normal distributions: human heights, test scores, measurement errors, and blood pressure readings. The Central Limit Theorem explains why – when you average many independent random variables, the result tends toward normality.

The Normal Distribution Formula

f(x) = (1 / σ√(2π)) x e^(-(x-μ)² / 2σ²)

Where μ = mean (center) and σ = standard deviation (spread)

68-95-99.7 Rule

68% within 1σ of mean
95% within 2σ of mean
99.7% within 3σ of mean

Z-Score Formula

z = (x - μ) / σ

Standardizes any normal to N(0,1)

Properties

Symmetric about mean
Mean = Median = Mode
Total area = 1

Worked Examples

Example 1: Standard Normal Probability

Find P(Z < 1.96) for standard normal

z = 1.96 (already standardized)

P(Z < 1.96) = 0.9750 = 97.5%

This is the critical value for 95% confidence!

Example 2: Test Scores

Test scores: μ = 75, σ = 10

What % scored below 85?

z = (85 - 75) / 10 = 1.0

P(X < 85) = P(Z < 1) = 0.8413 = 84.13%

Example 3: Finding Percentile

IQ scores: μ = 100, σ = 15

Find the 95th percentile

z for 95th percentile = 1.645

x = 100 + 1.645(15) = 124.675

95% of people score below 125

Example 4: Two-Tailed Probability

Find P(-1 < Z < 1)

P(Z < 1) = 0.8413

P(Z < -1) = 0.1587

P(-1 < Z < 1) = 0.8413 - 0.1587 = 0.6826

About 68% – matches the empirical rule!

Quick Fact

The normal distribution was first described by Abraham de Moivre in 1733 as an approximation to the binomial distribution. Carl Friedrich Gauss later used it to analyze astronomical data, which is why it's sometimes called the Gaussian distribution. The bell curve shape appears throughout nature due to the Central Limit Theorem.

Frequently Asked Questions

What does the Z-score tell me?

The Z-score tells you how many standard deviations a value is from the mean. Z = 0 is at the mean, Z = 1 is one standard deviation above, Z = -2 is two below. It standardizes any normal distribution to the standard normal N(0,1).

Why is the total area under the curve equal to 1?

The area represents probability. Since something must happen (100% certainty), the total area equals 1 (or 100%). The area between any two points gives the probability of a value falling in that range.

When can I use the normal distribution?

Use it when data is symmetric and bell-shaped, or when working with sample means (Central Limit Theorem). Many statistical tests assume normality. For small samples or skewed data, consider other distributions.

What's the difference between PDF and CDF?

PDF (probability density function) gives the height of the curve at a point. CDF (cumulative distribution function) gives the area under the curve up to that point – the probability of being less than or equal to that value.

How do I find the probability between two values?

Find the CDF for each value, then subtract: P(a < X < b) = P(X < b) - P(X < a). For example, P(-1 < Z < 1) = 0.8413 - 0.1587 = 0.6826.

What are critical values?

Critical values are Z-scores that correspond to specific tail probabilities. Common ones: Z = 1.96 for 95% confidence (two-tailed), Z = 1.645 for 95% (one-tailed), Z = 2.576 for 99% confidence.

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